Illinois Journal of Mathematics

Remarks on the quantum Bohr compactification

Matthew Daws

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The category of locally compact quantum groups can be described as either Hopf $*$-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how Sołtan’s quantum Bohr compactification can be used to construct a “compactification” in this category. Depending on the viewpoint, different C$^{*}$-algebraic compact quantum groups are produced, but the underlying Hopf $*$-algebras are always, canonically, the same. We show that a complicated range of behaviours, with C$^{*}$-completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of Sołtan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in $L^{\infty}(\mathbb{G})$, involving the antipode, which allows one to compute the Hopf $*$-algebra of the compactification of $\mathbb{G} $; we later study when the antipode assumption can be dropped. In the cocommutative case, we make a detailed study of Runde’s notion of a completely almost periodic functional—with a slight strengthening, we show that for [SIN] groups this does recover the Bohr compactification of $\hat{G}$.

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Illinois J. Math., Volume 57, Number 4 (2013), 1131-1171.

First available in Project Euclid: 1 December 2014

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Primary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 46L89: Other "noncommutative" mathematics based on C-algebra theory [See also 58B32, 58B34, 58J22]
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 43A20: $L^1$-algebras on groups, semigroups, etc. 43A95: Categorical methods [See also 46Mxx] 47L25: Operator spaces (= matricially normed spaces) [See also 46L07]


Daws, Matthew. Remarks on the quantum Bohr compactification. Illinois J. Math. 57 (2013), no. 4, 1131--1171. doi:10.1215/ijm/1417442565.

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